The so called consistent histories interpretation is claimed to be a correction of the Copenhagen Interpretation. One of its aim, as much as I can see is to show that observers don't have any special role and a history is represented as a sequence of projection
operators in the Heisenberg picture. It has been convincingly shown that due to environmental decoherence the density matrix will be almost diagonalized within a short time (expressed in terms of a certain basis of interest). It will smell just like a "collapse" of the wave function.

However, if we want to see the whole picture, we see that superposition still exists in the entangled whole. My question is whether this interpretation is actually a "many worlds interpretation" in disguise? If not, then why not?

I know your question is more philosophical, i.e. whether there is some theorem showing the two languages are equivalent, but on the practical side, the consistent histories approach gave rise to the first estimated for the time until loss of coherence. This was refined in the work of many. Does the many worlds interpretation do that, even today?
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sigoldberg1Jan 25 '11 at 18:13

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Sigoldberg, I completely agree with your text in between the lines. MW is more a psychological way to imagine things rather than a framework to calculate or predict things quantitatively. Consistent Histories can do the calculation, and it was the framework where decoherence was properly formulated - quantitatively. Still, the people behind Consistent Histories equally credit Hugh Everett - the father of many worlds - for the "history-based" approach that was important to create Consistent Histories.
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Luboš MotlJan 25 '11 at 18:28

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However, the top people behind CH rightfully admit that the Consistent Histories interpretation is just a refinement of the Copenhagen probabilistic interpretation. After all, it's very likely that Niels Bohr himself implicitly understood decoherence - he was just unable to formulate his understanding in a comprehensible a sharp way, something that occurred to Niels many times. ;-)
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Luboš MotlJan 25 '11 at 18:29

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Dear sb1, the top experts behind the Consistent Histories often credit Hugh Everett - the father of the many-worlds interpretation - with making the foundational quantum physics community focus on the "histories" as the basic entity whose probability quantum mechanics predicts. However, this is where the overlap ends.

People behind Consistent Histories usually admit that their interpretation - my favorite one - is just a refinement of the probabilistic Copenhagen interpretation. Nothing essential has changed; the predictions are still fundamentally probabilistic. Consistent Histories is the framework that incorporated the explanations of decoherence - the key process that calculates the boundary of the classical and quantum world - as the first one (and maybe still only one). Many-worlds interpretation is just a semi-popular psychological framework to think about quantum mechanics - and it hasn't been useful to do any actual, new calculations. One doesn't really know how to extract the numerical values of the probabilities from the many worlds, at least not in a way that would tell us more than any other interpretations.

Niels Bohr is said to have understood the essence of decoherence but he was unable to transform his understanding into comprehensible explanations or equations.

Decoherence is not quite the collapse of a wave function

Also, you say that there is a similarity between decoherence and the collapse of the wave function. Indeed, decoherence is what replaces the flawed ideas about the "collapse" and it is the key process that creates "classical objects" for which our non-quantum intuition is OK. However, all the details are completely different.

First of all, decoherence can be fully derived from the standard equations of quantum mechanics - from evolution combined with tracing over the environmental degrees of freedom. It doesn't try to distort the laws of quantum mechanics in any way: it just shows a generic and previously unappreciated consequence of these equations that is responsible for explaining lots of old puzzles.

On the contrary, the idea of a "collapse of the wave function" always tries to incorrectly treat the wave function as a real classical wave. For it to "physically" collapse, there must be new mechanisms and new terms added to the equations that make it collapse at a certain point. Needless to say, all these mechanisms are fictitious and all the mechanisms responsible for such a huge effect can be easily excluded. Quantum mechanics doesn't imply anything of the sort and the tests of quantum mechanics are not compatible with anything of the sort.

An important technical difference between decoherence and the collapse of the wave function is that decoherence actually doesn't decide which result will be measured. It just diagonalizes the density matrix to $\mbox{diag}(p_1,p_2,\dots,p_N)$ where $p_i$ then play the role of the probabilities that the individual preferred basis vectors will be detected. Because the information about the relative phases is getting quickly lost, the off-diagonal elements rapidly converge to zero. But decoherence never transforms the density matrix with the many $p$'s to something like $(0,0,1,0,\dots,0)$. Never, ever. Quantum mechanics remains the only player here and its predictions remain and will always stay probabilistic. One can never and one will never restore determinism and decoherence doesn't try to do anything of the sort.

I agree, decoherence helps us understanding quantum theory in much clear light by eliminating non unitary collapse from QT. But if unitarity is preserved at all times although no significant interference will take place due to dephasing, all the possible outcomes are still there in the entangled whole (system under consideration plus environment). Doesn't that bring us back to the essential idea of the many worlds?
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user1355Jan 26 '11 at 2:20

Dear sb1, many worlds is just a concept you may imagine but you don't have to. Whether you imagine them or not, it's still true that quantum mechanics can only make probabilistic predictions and there is no "mechanism" behind the selection of the random answers. I choose not to imagine the many worlds although I think in the Consistent Histories framework - at least whenever I am forced to go beyond the basic methodology of QM which is not often. That proves that the answer to your last question is "No, it doesn't bring us back."
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Luboš MotlJan 26 '11 at 6:27

No, in the many worlds interpretation, every parallel universe is real, but in consistent histories, once you choose your projection operators, only one possibility is real, and all the others are imaginary. This makes consistent histories a lot more like Bohmian mechanics with the world the Bohmian particle sits in being more real than the rest. Why should one world be more real than the others? There is no reason. To copies of you living in a parallel world, they are more real than you are.